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To alleviate the suboptimal performance of belief propagation (BP) decoding of short low-density parity-check (LDPC) codes, a plethora of improved decoding algorithms has been proposed over the last two decades. Many of these methods can be described using the same general framework, which we call ensemble decoding: A set of independent constituent decoders works in parallel on the received sequence, each proposing a codeword candidate. From this list, the maximum likelihood (ML) decision is designated as the decoder output. In this paper, we qualitatively and quantitatively compare different realizations of the ensemble decoder, namely multiple-bases belief propagation (MBBP), automorphism ensemble decoding (AED), scheduling ensemble decoding (SED), noise-aided ensemble decoding (NED) and saturated belief propagation (SBP). While all algorithms can provide gains over traditional BP decoding, ensemble methods that exploit the code structure, such as MBBP and AED, typically show greater performance improvements., Comment: 6 pages, 7 figures, submitted to IEEE for possible publication

In this paper, we propose a data-driven algorithm to design rate- and length-flexible polar codes. While the algorithm is very general, a particularly appealing use case is the design of codes for automorphism ensemble decoding (AED), a promising decoding algorithm for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC) applications. To this end, theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED. Using the proposed algorithms, such a flexible polar code design for automorphism ensemble successive cancellation (SC) decoding is constructed, outperforming existing code designs for AED and also the 5G polar code under cyclic redundancy check (CRC)-aided successive cancellation list (SCL) decoding., Comment: 6 pages, 2 figures, submitted to IEEE for possible publication

For asynchronous transmission of short blocks, preambles for packet detection contribute a non-negligible overhead. To reduce the required preamble length, joint detection and decoding (JDD) techniques have been proposed that additionally utilize the payload part of the packet for detection. In this paper, we analyze two instances of JDD, namely hybrid preamble and energy detection (HyPED) and decoder-aided detection (DAD). While HyPED combines the preamble with energy detection for the payload, DAD also uses the output of a channel decoder. For these systems, we propose novel achievability and converse bounds for the rates over the binary-input additive white Gaussian noise (BI-AWGN) channel. Moreover, we derive a general bound on the required blocklength for JDD. Both the theoretical bound and the simulation of practical codebooks show that the rate of DAD quickly approaches that of synchronous transmission., Comment: 6 Pages, 3 Figures, submitted to IEEE for possible publication

This paper proposes to use graph neural networks (GNNs) for equalization, that can also be used to perform joint equalization and decoding (JED). For equalization, the GNN is build upon the factor graph representations of the channel, while for JED, the factor graph is expanded by the Tanner graph of the parity-check matrix (PCM) of the channel code, sharing the variable nodes (VNs). A particularly advantageous property of the GNN is the robustness against cycles in the factor graphs which is the main problem for belief propagation (BP)-based equalization. As a result of having a fully deep learning-based receiver, joint optimization instead of individual optimization of the components is enabled, so-called end-to-end learning. Furthermore, we propose a parallel flooding schedule that further reduces the latency, which turns out to improve also the error correcting performance. The proposed approach is analyzed and compared to state-of-the-art baselines in terms of error correcting capability and latency. At a fixed low latency, the flooding GNN for JED demonstrates a gain of 2.25 dB in bit error rate (BER) compared to an iterative Bahl--Cock--Jelinek--Raviv (BCJR)-BP baseline., Comment: Submitted to ISIT 2024

Row-merged polar codes are a family of pre-transformed polar codes (PTPCs) with little precoding overhead. Providing an improved distance spectrum over plain polar codes, they are capable to perform close to the finite-length capacity bounds. However, there is still a lack of efficient design procedures for row-merged polar codes. Using novel weight enumeration algorithms with low computational complexity, we propose a design methodology for row-merged polar codes that directly considers their minimum distance properties. The codes significantly outperform state-of-the-art cyclic redundancy check (CRC)-aided polar codes under successive cancellation list (SCL) decoding in error-correction performance. Furthermore, we present fast simplified successive cancellation list (Fast-SSCL) decoding of PTPCs, based on which we derive a high-throughput, unrolled architecture template for fully pipelined decoders. Implementation results of SCL decoders for row-merged polar codes in a 12 nm technology additionally demonstrate the superiority of these codes with respect to the implementation costs, compared to state-of-the-art reference decoder implementations., Comment: 13 pages, 7 figures, Submitted to IEEE for possible publication

Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this work, we propose an efficient algorithm for determining the number of minimum weight codewords of general PTPCs that eliminates all redundant visits to nodes of the search tree, thus reducing the computational complexity typically by several orders of magnitude compared to state-of-the-art algorithms. This reduction in complexity allows, for the first time, the minimum distance properties to be directly considered in the code design of PTPCs. The algorithm is demonstrated for randomly pre-transformed Reed-Muller (RM) codes and polarization-adjusted convolutional (PAC) codes. Furthermore, we design optimal polynomials for PAC codes with this algorithm, minimizing the number of minimum weight codewords., Comment: 8 pages, 4 figures, extended version of the CISS 2024 paper

The discovery of suitable automorphisms of polar codes gained a lot of attention by applying them in Automorphism Ensemble Decoding (AED) to improve the error-correction performance, especially for short block lengths. This paper introduces Successive Cancellation Automorphism List (SCAL) decoding of polar codes as a novel application of automorphisms in advanced Successive Cancellation List (SCL) decoding. Initialized with L permutations sampled from the automorphism group, a superposition of different noise realizations and path splitting takes place inside the decoder. In this way, the SCAL decoder automatically adapts to the channel conditions and outperforms the error-correction performance of conventional SCL decoding and AED. For a polar code of length 128, SCAL performs near Maximum Likelihood (ML) decoding with L=8, in contrast to M=16 needed decoder cores in AED. Application-Specific Integrated Circuit (ASIC) implementations in a 12 nm technology show that high-throughput, pipelined SCAL decoders outperform AED in terms of energy efficiency and power density, and SCL decoders additionally in area efficiency., Comment: 5 pages, 5 figures, submitted to IEEE for possible publication

Isolated training with Gaussian priors (TGP) of the component autoencoders of turbo-autoencoder architectures enables faster, more consistent training and better generalization to arbitrary decoding iterations than training based on deep unfolding. We propose fitting the components via extrinsic information transfer (EXIT) charts to a desired behavior which enables scaling to larger message lengths ($k \approx 1000$) while retaining competitive performance. To the best of our knowledge, this is the first autoencoder that performs close to classical codes in this regime. Although the binary cross-entropy (BCE) loss function optimizes the bit error rate (BER) of the components, the design via EXIT charts enables to focus on the block error rate (BLER). In serially concatenated systems the component-wise TGP approach is well known for inner components with a fixed outer binary interface, e.g., a learned inner code or equalizer, with an outer binary error correcting code. In this paper we extend the component training to structures with an inner and outer autoencoder, where we propose a new 1-bit quantization strategy for the encoder outputs based on the underlying communication problem. Finally, we discuss the model complexity of the learned components during design time (training) and inference and show that the number of weights in the encoder can be reduced by 99.96 %., Comment: Submitted to ISTC 2023,5 pages

For short-packet, low-latency communications over random access channels, piloting overhead significantly reduces spectral efficiency. Therefore, pilotless systems recently gained attraction. While blind phase estimation algorithms such as Viterbi-Viterbi Phase Estimation (VVPE) can correct a phase offset using only payload symbols, a phase ambiguity remains. We first show that the remaining phase rotations in a polar coded quadrature amplitude modulation (QAM) transmission with gray labeling are combinations of bit-flips and automorphisms. Therefore, the decoder is equivariant to such phase rotations and, by smartly selecting the frozen bits, one can jointly decode and resolve the phase ambiguity, without the need for pilot symbols or an outer code. Our proposed system outperforms pilot-assisted transmissions by up to 0.8 dB and 2 dB for quaternary phase shift keying (QPSK) and 16-QAM, respectively., Comment: 5 pages, 6 figures, submitted to IEEE for possible publication

Recently, automorphism ensemble decoding (AED) has drawn research interest as a more computationally efficient alternative to successive cancellation list (SCL) decoding of polar codes. Although AED has demonstrated superior performance for specific code parameters, a flexible code design that can accommodate varying code rates does not yet exist. This work proposes a theoretical framework for constructing rate-compatible polar codes with a prescribed automorphism group, which is a key requirement for AED. We first prove that a one-bit granular sequence with useful automorphisms cannot exist. However, by allowing larger steps in the code dimension, flexible code sequences can be constructed. An explicit synthetic channel ranking based on the $\beta$-expansion is then proposed to ensure that all constructed codes possess the desired symmetries. Simulation results, covering a broad range of code dimensions and blocklengths, show a performance comparable to that of 5G polar codes under cyclic redundancy check (CRC)-aided SCL decoding, however, with lower complexity., Comment: 5 pages, 2 figures, submitted to IEEE for possible publication

The URLLC scenario in the upcoming 6G standard requires low latency and ultra reliable transmission, i.e., error correction towards ML performance. Achieving near-ML performance is very challenging especially for short block lengths. Polar codes are a promising candidate and already part of the 5G standard. The Successive Cancellation List (SCL) decoding algorithm provides very good error correction performance but at the cost of high computational decoding complexity resulting in large latency and low area and energy efficiency. Recently, Automorphism Ensemble Decoding (AED) gained a lot of attention to improve the error correction capability. In contrast to SCL, AED performs several low-complexity (e.g., SC) decoding in parallel. However, it is an open question whether AED can compete with sophisticated SCL decoders, especially from an implementation perspective in state of the art silicon technologies. In this paper we present an elaborated AED architecture that uses an advanced path metric based candidate selection to reduce the implementation complexity and compare it to state of the art SCL decoders in a 12nm FinFET technology. Our AED implementation outperform state of the art SCL decoders by up to 4.4x in latency, 8.9x in area efficiency and 4.6x in energy efficiency, while providing the same or even better error correction performance., Comment: 6 pages, International ITG 26th Workshop on Smart Antennas (WSA) and 13th Conference on Systems, Communications, and Coding (SCC), 2023

For improving short-length codes, we demonstrate that classic decoders can also be used with real-valued, neural encoders, i.e., deep-learning based codeword sequence generators. Here, the classical decoder can be a valuable tool to gain insights into these neural codes and shed light on weaknesses. Specifically, the turbo-autoencoder is a recently developed channel coding scheme where both encoder and decoder are replaced by neural networks. We first show that the limited receptive field of convolutional neural network (CNN)-based codes enables the application of the BCJR algorithm to optimally decode them with feasible computational complexity. These maximum a posteriori (MAP) component decoders then are used to form classical (iterative) turbo decoders for parallel or serially concatenated CNN encoders, offering a close-to-maximum likelihood (ML) decoding of the learned codes. To the best of our knowledge, this is the first time that a classical decoding algorithm is applied to a non-trivial, real-valued neural code. Furthermore, as the BCJR algorithm is fully differentiable, it is possible to train, or fine-tune, the neural encoder in an end-to-end fashion., Comment: WSA/SCC 23, 6 pages

It is well known that to fulfill their full potential, the design of polar codes must be tailored to their intended decoding algorithm. While for successive cancellation (SC) decoding, information theoretically optimal constructions are available, the code design for other decoding algorithms (such as belief propagation (BP) decoding) can only be optimized using extensive Monte Carlo simulations. We propose to view the design process of polar codes as a graph search problem and thereby approaching it more systematically. Based on this formalism, the design-time complexity can be significantly reduced compared to state-of-the-art Genetic Algorithm (GenAlg) and deep learning-based design algorithms. Moreover, sequences of rate-compatible polar codes can be efficiently found. Finally, we analyze both the complexity of the proposed algorithm and the error-rate performance of the constructed codes., Comment: 5 pages, 8 figures, accepted to the 2022 Asilomar Conference on Signals, Systems, and Computers

Finding optimal message quantization is a key requirement for low complexity belief propagation (BP) decoding. To this end, we propose a floating-point surrogate model that imitates quantization effects as additions of uniform noise, whose amplitudes are trainable variables. We verify that the surrogate model closely matches the behavior of a fixed-point implementation and propose a hand-crafted loss function to realize a trade-off between complexity and error-rate performance. A deep learning-based method is then applied to optimize the message bitwidths. Moreover, we show that parameter sharing can both ensure implementation-friendly solutions and results in faster training convergence than independent parameters. We provide simulation results for 5G low-density parity-check (LDPC) codes and report an error-rate performance within 0.2 dB of floating-point decoding at an average message quantization bitwidth of 3.1 bits. In addition, we show that the learned bitwidths also generalize to other code rates and channels., Comment: 6 Pages, 11 Figures, submitted to IEEE for possible publication

Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed algorithm outperforms the previously best performing iterative CRC-aided belief propagation list (CA-BPL) decoder both in error-rate performance and decoding latency., Comment: 6 pages, submitted to IEEE for possible publication

We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed method is demonstrated using LDPC codes from the CCSDS, 802.11n and 5G standards and produces gains of 0.2 to 0.3 dB over conventional BP decoding. Compared to the similarly performing saturated BP (SBP), the proposed algorithm reduces the average decoding latency by more than eight times., Comment: 5 pages, submitted to IEEE for possible publication

Reed-Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a belief propagation (BP) decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) using polar BP as constituent decoders. We provide extensive error-rate performance simulations and compare our results to existing decoding schemes. We report a near-ML performance for the RM(3,7)-code (e.g., 0.05 dB away from the ML bound at BLER of $10^{-4}$) at a competitive computational cost. To the best of our knowledge, our proposed decoder achieves the best performance of all iterative RM decoders presented thus far., Comment: 5 pages, accepted for publication at the International Symposium on Topics in Coding 2021 (ISTC), Sep. 2021. Short version of arXiv:2012.07635

The automorphism group of a code is the set of permutations of the codeword symbols that map the whole code onto itself. For polar codes, only a part of the automorphism group was known, namely the lower-triangular affine group (LTA), which is solely based upon the partial order of the code's synthetic channels. Depending on the design, however, polar codes can have a richer set of automorphisms. In this paper, we extend the LTA to a larger subgroup of the general affine group (GA), namely the block lower-triangular affine group (BLTA) and show that it is contained in the automorphism group of polar codes. Furthermore, we provide a low complexity algorithm for finding this group for a given information/frozen set and determining its size. Most importantly, we apply these findings in automorphism-based decoding of polar codes and report a comparable error-rate performance to that of successive cancellation list (SCL) decoding with significantly lower complexity., Comment: 7 Pages, extended version of ISIT 2021 submission

Reed-Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of $10^{-3}$) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes., Comment: Accepted for Publication in IEEE Transactions on Communications

We consider the usage of finite-length polar codes for the Gaussian multiple access channel (GMAC) with a finite number of users. Based on the interleave-division multipleaccess (IDMA) concept, we implement an iterative detection and decoding non-orthogonal multiple access (NOMA) receiver that benefits from a low complexity, while scaling (almost) linearly with the amount of active users. We further show the conceptual simplicity of the belief propagation (BP)-based decoder in a step-by-step illustration of its construction. Beyond its conceptual simplicity, this approach benefits from an improved performance when compared to some recent work tackling the same problem, namely the setup of finite-length forward errorcorrection (FEC) codes for finite-number of users. We consider the 5th generation mobile communication (5G) polar code with a block length $N = 512$ applied to both a two-user and a four-user GMAC scenario with a sum-rate of $R_{sum} = 0.5$ and $R_{sum} = 1$, respectively. Simulation results show that a BP-based soft interference cancellation (SoIC) receiver outperforms a joint successive cancellation (JSC) scheme. Finally, we investigate the effect of a concatenated repetition code which suggests that alternative polar code design rules are required in multi-user scenarios., Comment: Asilomar2020

Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopping condition (i.e., for error-detection), we also aim to benefit from the error-correction capabilities of the outer CRC code. For this, we develop two distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv (BCJR)-based approach and a sum product algorithm (SPA)-based approach. Further, an optimized selection of permuted factor graphs is analyzed and shown to reduce the decoding complexity significantly. Finally, we benchmark the proposed CRC-aided belief propagation list (CA-BPL) to state-of-the-art 5G polar codes under CA-SCL decoding and, thereby, showcase an error-rate performance not just close to the CA-SCL but also close to the maximum likelihood (ML) bound as estimated by ordered statistic decoding (OSD)., Comment: Submitted to IEEE for possible publication